The sum of all possible values of θ ∈ [ − π , 2 π ] , for which 1 − 2 i c o s θ 1 + i c o s θ is purely imaginary, is equal

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The sum of all possible values of θ ∈ [ − π , 2 π ] , for which 1 − 2 i c o s θ 1 + i c o s θ ​ is purely imaginary, is equal

TestHub · Accepted Answer

​ Z = 1 − 2 i cos θ 1 + i cos θ ​ Z = − Z ˉ ⇒ 1 − 2 i cos θ 1 + i cos θ ​ = − ( 1 − 2 i cos θ 1 + i cos θ ​ ​ ) ( 1 + i cos θ ) ( 1 − 2 i cos θ ​ ) = − ( 1 − 2 i cos θ ) ( 1 + i cos θ ​ ) ( 1 + i cos θ ) ( 1 + 2 i cos θ ) = − ( 1 − 2 i cos θ ) ( 1 − i cos θ ) 1 + 3 i cos θ − 2 cos 2 θ = − ( 1 − 3 i cos θ − 2 cos 2 θ ) 2 − 4 cos 2 θ = 0 ⇒ cos 2 θ = 2 1 ​ ⇒ θ = − 4 π ​ , − 4 3 π ​ , 4 π ​ , 4 3 π ​ , 4 5 π ​ , 4 7 π ​ sum = 3 π ​

Mathematics - Complex Number Question with Solution | TestHub

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